Primary-geodesic least surface shapes for predetermined ground plots and functions

ABSTRACT

A means to create the shape for a structure that holds the most space under the least area for any predetermined polygonal ground plot. The ground plot is cut into a sheet material as an open window, called a plot-frame  2 . The plot-frame  2  could also be created on the ground itself. A membrane  1  is placed over the plot-frame  2  and a pressure differential is created on both sides of the membrane  1 . Functions that take on polygonal shapes are placed on the membrane  1  that would be entryways; connection planes and other flat polygons are achieved by means of constraints  4.  The final static display shape of the membrane  1  shows the shape of the intended structure above ground.

This invention is based on my provisional application No. 60/528158,dated Dec. 9, 2003

TECHNICAL FIELD

This artwork relates to all least surface space enclosing structuresover predetermined ground plots with functions placed on the surfacesuch as entryways, connection planes, solar panels and the like. Suchartwork is called geodesic domes and space enclosing structures. Thereis an infinite set of shapes other than the sphere that holds space asleast surface shape, that was said could be geodesic.

BACKGROUND ART

Within the history of man the first geodesic-like sphere has been underthe paws of the Guardian Lions that have been placed outside the templesand gates of China at least 400 years ago. Like all of the currentgeodesic structures, it seems to resemble a polyhedral made of trianglesthat resemble a sphere. However, having a geodesic like sphere is notthe same as a space enclosing structure.

Next there is a true geodesic dome created by Dr. Walter Bauersfeld in1922 in Jena, Germany, used for a planetarium-roof that was theembodiment of his work. This structure was made from a dividedicsoahedron cast to a sphere to allow his projection devices to remainin focus. It can be assumed he projected rays to the junctures of thematerial he used as spherical chords that was his framework for this,from a central point. It could also be assumed he didn't see the domestructure which he created, as the roof of the Carl Zeiss optical works,as a single structure. His concern may have just been to have a propersurface for his projectors to function correctly, and the dome itselfwas perhaps overlooked as a single work.

Richard Buckminster Fuller's first artwork on this subject in 1954, U.S.Pat. No. 2,682,235 made it singular and useful unto itself. It is whatFuller said a geodesic dome could be after this artwork, which mattersmost of all. Published within Synergetics in 1975, in section 703.01 hesays “Geodesic domes can be either symmetrically spherical, like abilliard ball, or asymmetrically spherical, like pears, caterpillars, orelephants.” He said also in section 703.03 “All geodesic domes aretensegrity structures whether or not the tension-compressiondifferentiations are visible to the observer”. “Tensegrity” is a shortform of the term “tensional integrity”. Within section 702.01 he alsosaid; “We have a mathematical phenomenon known as a geodesic. A geodesicis the most economical relationship between any two events”. If Fulleris to be given credit for the geodesic dome, the statements should becombined. To me they mean that it is the shape, and not the polyhedralsurface that represents what is a geodesic. I call such shapes “primarygeodesic surfaces” and the polyhedral representation of them as a“secondary geodesic” no matter the pattern of polygons. He removed thecommonly assumed mathematical rules that govern the current geodesicdomes and the polyhedral surface. There are no mathematical rules thatcould be applied to the surface of an elephant Fuller never explainedhow to create such geodesic surfaces, only that they would be geodesicdomes. This current artwork covers the creation of them.

Still looking at Fuller and the bulk of the remaining artwork, most showthe polyhedral representation of the sphere as a structure that iscalled a geodesic dome. Hannula with his U.S. Pat. No. 3,955,329 in 1974titled as “Hollow Structure” gives a curved set of lines to become ageodesic-like on the surface of the sphere. Also Herrmann U.S. Pat. No.6,295,785 with a pattern based on the octahedron and not the icsoahedronof Fullers first work U.S. Pat. No. 2,682,235. Within Herrmann's workcomes the voice of Yacoe U.S. Pat. No. 4,679,361 saying a geodesic is arepresentation of a sphere. Leonard Spunt U.S. Pat. No. 3,959,937,“Modular dome structure” in 1976, shows a sphere dome done in circlesthat is also a polygon much like the triangle, depending on yourschooling in geometry. Spunt shows us that the circles could be fromcones, all with the tips concentric to the sphere and the axis of each aray from that center. Circles would come from the intersection of thecones and the sphere. All the commonalities of each of the many geodesicand space-enclosing artworks show a polyhedral representation of thesurface of the sphere as a shell of polygons. This leads us to thinkthat there could be more geodesic patents than anyone might care toguess at, that could be copied to a sphere. Also some that look verygeodesic seem to just be called “structures”, making the line betweenthe two types very blurred. For all the effort placed into findingcommon sets of points between the sphere and some solids, rays and conescast from the center of the sphere, none are able to explain theelephant or caterpillar. Maybe it's time to stop finding all themathematical sets of points common to the sphere and whatever method orsolid, used to divide a sphere into a geodesic. Fuller told us that thesphere is a geodesic if one can see a polygonal pattern or not. Perhapsdefining the geodesic dome as a polyhedral representation of just asurface is incorrect. Maybe it should be looked at as least surfacerelationships that nature can produce as geodesic structures, and whatFuller said they can be. This current artwork shows us how to do justthat. It is also able to reproduce with this current artwork, many pastgeodesic polyhedral surfaces with just 3 parts.

Of all the works remaining on space enclosing structures there are threepeople that need to be noted. The first is Helmut Bergman U.S. Pat. No.4,258,513 showing that a structure can have rectangles providing afunction on the surface for solar collectors by providing a place tomount them. Also from Helmut Bergman there is U.S. Pat. No. 4,364,207that can allow for a changing ground plot titled “Extended SpaceEnclosing Structure”; it shows a somewhat variable ground plot. Hisworks are creative because they show that such structures can be closeto the geodesic sphere and have the surface changed to also includerectangles, for some required function such as for solar-panels, doorsor clustering. His creations are also based on the icsoahedron as aremany others, but one has to look closely. However creative this fallsshort of showing a least surface over any ground plot. The extendedstructure is limited to an elongated circle only. However it is the ideaof polygons other than triangles as an intended function that makes mycurrent artwork more valuable. I have to extend a full measure of creditto Bergman for showing us this. I hope to have fully exploited histeaching with this present artwork.

The last two remaining artworks worth mentioning starts with David BSouth U.S. Pat. No. 4,155,967 in 1979 and U.S. Pat. No. 4,324,074 in1982. He inflates an inelastic membrane that is formed to bespherical-like; because of the manner in which it was made and used. Hissystem relies on the pneumatic pressure within it to become stiff. Laterin 1999 with U.S. Pat. No. 5,918,438 he told us this again, when hefound the need to place a net over the membrane to help it retain thatshape. Because this is a sphere based structure, I consider it a primarygeodesic, because there can be seen no “tension-compressiondifferentiations” as noted by Fuller above. Unfortunately, with the workof David South, the shape of the structure is predetermined at the timeof manufacture of that membrane. It only becomes rigid under pressureand is unable to produce a least surface area for the space it bindsbetween it and the ground when inflated. It is unable to conform to anyground plot. Making a membrane in that manner for a different groundplot, as a least surface above that ground, would require a guess atbest. He has not been able to teach us the means to find the othershapes Fuller talked about. This current artwork has no such shortcomings.

Along side and before David South there is the work that can be seen onthe internet at www.binisystems.com/binisystems.html. Here Dante N Binishows us a system much like that used by David South, but the membranehas the ability to be elastic and can be stopped at any level ofinflation. However as the video shows within that web page, connectionsto the planes are made by cutting away the surface of one dome to comeinto contact with the next. The perimeter of the second dome has toencroach into the perimeter of the first. This tends to limit when andwhere such a system can be employed, and requires virgin ground. Thissystem would be unable to maintain the least surface with the connectionplane in place as a flat polygon. Such would be the case if the dome hadto connect to a flat surface from an existing structure. Even if thislesson from Dante Bini is most close to this current artwork, it can'tshow us how to implement the polygonal functions that Helmut Bergman hasgiven us. He did not explain how to produce the other asymmetricallyspherical shapes Fuller told us about. This current artwork takes careof the need for virgin ground and cutting away sections to connect thedome to other structures, and any encroachments into that structure.

To account for the geodesic dome as to what Fuller said it can be it maybe time to let go of the idea of projecting points, lines and circles tothe sphere. The sphere is only a mathematical real world model of aleast surface shape for the volume it holds, and the best at that. Itmay be that the sphere has been used almost exclusively until now,because its math is relatively easy to work, the points easy to produce.This current artwork removes the math of the surface and allows us touse a simpler means to create even more complex geodesic surfaces withfunctions on its surface.

BRIEF SUMMERY

This current artwork was created to confirm an equation I found for ageodesic dome with a square foot print to the ground. The shape that themembrane 1 displayed, matched the equation well. It did show a squaregeodesic dome can have an area only 3.0-4.5 percent above the area of asphere that holds twice the volume that is displaced by the membrane 1.In other words twice the area of the displaced membrane 1, is only3.0-4.5 percent above the area of a sphere that can hold twice the spacedisplaced by the membrane 1. Also I didn't know that Fuller had saidthey could be anything but spherical at that time. After it came to myattention what Fuller said geodesic domes could be, it was still manyweeks later that I realized what I had on my desk. I also have to credita math book that was resting on the surface of the membrane 1, forshowing me the membrane 1 is self-correcting.

The system works because of the nature of the membrane 1 and thepressure across its two surfaces. The tension in the membrane 1 causedby making it larger in area by that force, will act to return it to aleast area. In turn the pressure that caused the change in area wants toexpand in all directions. The net result is that the membrane 1 shows astatic display of the two opposing forces in balance. The membrane 1shows the least area for the amount of space it has displaced. If thepattern cut into the plot-frame 2 happens to be the similar polygonalshape of some predetermined ground plot; the membrane 1 shows all leastsurface areas for any amount of space that could be above thatground-plot, regardless of target height or the amount of space withinthe final structure. If the amount of space that is bound by themembrane 1 and the membrane 1 itself become very large, it also becomesvery spherical. Because of this action when made massive in size, itshows clearly that the membrane 1 is spherically packing the pressureagent. That tells us the membrane 1 is always seeking a least area. Muchas a small drop of water free from outside forces would. The reductionin area is a natural event for such systems.

With the complexity of the math I used to create this shape, themembrane 1 shows the more correct result. It shows surfaces forequations that may never be found. The membrane 1 allows us to map andcollect data for the shape it shows, so it can be reproduced in scale,or as the final size of the structure. All solutions that the membrane 1shows are least surface shapes. I would call such shapes“primary-geodesic” shapes. Expanding the membrane 1 from the open windowof the plot-frame 2 shows one side of the asymmetrically sphericalshapes that Fuller said are geodesic. The second half would be themirror reflection of the membrane 1 past the edge of the window cut intothe plot-frame 2. The primary shape maybe represented as a polyhedral,and than becomes what I call a secondary geodesic. The math of thesurface is no longer required. The collection of data points that makeup the surface is all that is necessary, or as already noted, thesurface itself.

It has to be the nature of that surface that the membrane 1 shows, whichbonds any amount of space, how that surface is shaped, and the functionseach surface has imposed on itself; that makes something geodesic. Justas it is the surface of the pear, elephant, and sphere, that allows themto be geodesic. All self seeking least areas are therefore geodesic. Themathematical rules used to explain the geodesic sphere will not work forall the other asymmetrically spherical cases. As nature shows us themost well-fed elephant is the most spherical, that too is a leastsurface seeking system.

Most of all the past artwork on geodesic structures can be reproducedwith this current one by making use of an appropriate plot-frame 2 and asingle constraint 4. The constraint 4 might look something like asea-urchin, with its tips in all the right places. With the membrane 1over this and the correct pressure differential across its face, eventhe polygonal surfaces that comprise the polyhedral geodesic dome can berecreated. By adding/removing, or deflecting the end point of any arm onthe constraint, any number of geodesic surfaces can be produced with oneplot-frame 2. Considering that there can be an infinite number ofconstraints and plot-frames 2, in any combination; the practical use ofthis invention comes clear.

Looking at the membrane 1 and its reflected surface and area past theplot-frame 2 there are some things worth noting. All secondary-geodesicsurfaces that are polyhedral representations of the primary-geodesicsurface presented by the membrane 1 have a higher surface/volume ratiothan the primary shape. All non hemispherical shapes produced by themembrane 1 have a higher surface/volume ratio than the sphere that holdsthe same volume as twice the one displaced by the membrane 1. As I havefound with a square window within the plot-frame 2 the amount that itssurface area to volume relationship is higher than the sphere, becomesunimportant when the functionally functionalism of the structure isconsidered. Each plot-frame has a null or dip in the surface area tovolume relationship as the amount of space displaced by the membranemoves from zero to infinity. Each different or geometric none-similarpolygon, cut into the plot-frame 2, has a different null number. Only acircle cut into a plot-frame 2 and a good quality membrane 1 willproduce a null value of zero. This happens only when the height of thedisplaced membrane 1 from the plot-frame 2 is equal to the radius of thecircle cut into the plot-frame 2. That is because one would have tocompare the S/V of the membrane to the S/V of the sphere as noted above.This is true for all none-constrained membranes 1. There can be nonegative null numbers. All constrained membranes 1 should have an S/Vratio above the null for that plot-frame 2.

Constraints only hold the membrane 1 to points/lines/planes that themembrane 1 may or may not reach on its own. These constraints may holdthe membrane 1 to a needed door frame size on the edge of a ground-plot.The remaining free area of the membrane 1 will still show the least areafor its displaced space no matter the amount. The only problems that canoccur come from the failure to make precise parts and the ability of themembrane 1 to maintain an even surface tension.

A good test for the quality of the membrane 1 is when it is expandedfrom a circular plot-frame 2 and how close it comes to a hemisphere whenthe height is equal to the radius of the circle. Because the shape ofthe sphere is known so well, the surface of the membrane 1 would bemapped and compared to that of a true hemisphere.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an expanded view of unit with a square within the plot-frame2;

FIG. 2 is an assembled view of FIG. 1;

FIG. 3 shows a static display of FIG. 2;

FIG. 4 the final shape of membrane 1 and the space inside;

FIG. 5 the space of FIG. 4 with a set of nodes for polygon frameworkconnections;

FIG. 6 one of any geodesic framework over the shape;

FIG. 7 a realization of Fullers Elephant with this system;

FIG. 8 shows one basic system of FIG. 1 unit with constraints for doors;

FIG. 9 gives shapes of the membrane 1 found in FIG. 8;

FIG. 10 is a more complex plot-frame 2 and simple wire constraint 4;

FIG. 11 is the primary geodesic surface shown in FIG. 10;

FIG. 12 gives one of many cluster systems that can be made with thisartwork.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 shows an expanded view of a plot-frame 2, with a square as theopen window that the membrane 1 will be pushed from. The shape cut intothe plot-frame 2 can be any closed shape. The plot-frame 2 is best madefrom a stiff, thin sheet material. It can also be that the ground servesas both the plot-frame 2 and the base 3. In that case the plot-frame 2would be a closed polygon, or cross section of any object upon theground.

The square in the plot-frame 2 of FIG. 1 could be a square predeterminedground-plot of any size. The base 3 allows us to seal the system andplace a pressure on one side of the membrane 1. The base 3 can be madefrom about anything that will hold up to the forces that will act on it.The amount of space between the membrane 1 and the base 3 form acontainment vessel. The pressure agent within that containment vesseland used to push on the membrane 1 can be any suitable and finelydivided substance. Some such elements could be but are not limited to;air, oil, none set plaster-like substance or water. The pressure canalso come from heat, chemicals or the removal of ambient air. It onlymatters that a pressure difference can be created and maintained. If anentry-port is required for some pressure agent, it is best locatedwithin the base 3 and within the open area of the plot-frame 2. Themembrane 1 is said to be one, that is able to become easily expanded,and will display an even surface tension when its surface is distorted.Most times a rubber-like compound will work best, but the membrane 1 isnot limited to this. A hot plastic in an almost liquid state, may alsobe employed, and allowed to cool and set hard.

FIG. 2 shows the assembled view of the unit. It shows a starting zerodisplaced volume across the open window of the plot-frame 2. The unit isassembled by any means that will hold the membrane 1 tightly toplot-frame 2 and to the containment vessel, that the pressure is foundwithin. Any means to bond this is usable, such as glue, screws, clampsand locking clips or a combination thereof. What ever the means tocombine and seal the unit is unimportant, so that has been omitted forclarity of view. At this stage the unit is assembled and ready. It canbe that the plot-frame 2 is also mounted to the side of a tank. Thevolume of space in that tank is disregarded, because the only concern iswith the amount of space that the membrane 1 displaces across theplot-frame 2. However if the plot-frame 2 was mounted to a tank largeenough, the pressure differential across the surfaces of the membrane 1could come from a reduction of pressure within that tank. In that casethe higher pressure would come from the ambient air. The membrane 1 willshow the same result no matter the direction of its displacement, aslong as it is free to do so.

In FIG. 3 the membrane 1 is displaced via some pressure increase betweenit and the base 3. The surface of the membrane 1 is larger. The amountof space that the membrane 1 moved past its rest state within the openwindow in the plot-frame 2, and the area upon the membrane 1 representthe size and shape of the intended final structure. It is showing theleast surface for that amount of space passed from that plot-frame 2,regardless of the volume of that space. It is at this stage that thesurface of the membrane 1 is mapped to within the three dimensions ofspace. The more detail in this mapping and analysis of the surface ofthe membrane 1, the better any reproduction will be. The final targetstructure will be similar to the shape of the membrane 1, be thatlarger, smaller or of equal size. Also the membrane it self might becomethe inner or outer shell of the structure it self; and then be removedand reused, or kept in place.

FIG. 4 shows the membrane 1 by itself or the shape of the space betweenthe membrane 1 and the base 3 if plaster was used as a pressure agentand allowed to harden. In such a manner as that, the shape can beremoved, copied and measured in more detail. It can also provide a scalecopy, showing the shape and volume of the structure. This would be ofuse if a model was required.

FIG. 5 shows some locations over FIG. 4 that could be nodes which wouldbe used to divide the surface with a set of chords producing apolyhedral representation of the surface in FIG. 4. The amount of nodesand the distances between them is all relative to the wishes of theperson that is to produce the final structure. One such wish could be tohave all the chords about equal. A different intention might be to havethe least waste from an unlimited pile of lumber made up of 8 footpieces. The larger the number of nodes the more the final structure willresemble the surface of the membrane 1. When the surface of FIG. 4 wasmapped in to three-dimensional spaces, so were the nodes. It is than upto the rules of analytical geometry to show us the distances and angles,or any other information that will aid us to be able to produce thepolyhedral surface. The polyhedral surface for FIG. 5's nodes can beseen in FIG. 6.

At this stage all the details have been covered to produce the elephantor any other object Fuller talked about. The only change is to thewindow cut into the plot-frame 2. In FIG. 7A, there is a plot-frame 2with the shape of the side view, or horizontally lit shadow of anelephant upon a wall cut into it. The base 3 is omitted or assumed to beunder the plot-frame 2 for clarity of view. FIG. 7B shows what themembrane 1 would be like when reflected past the open window cut intothe plot-frame 2 in FIG. 7A. For the view of 7B to be clear the base isomitted. What can be seen in FIG. 7B is that any increases or decreaseswith the pressure differential on the sides of the membrane 1 will showan elephant that could be better or lesser fed, much as nature would inthe real world. To represent the caterpillar, I would cut the top viewof one as seen on the ground into the plot-frame 2. The result would bea tunnel like shape on the membrane 1. This approach would even work ifthe caterpillar was bent, as if it was in the middle of making a turn.For the pear one would be cut from stem to navel, a copy thatcross-section would be cut into the plot-frame 2 as an open window. Ifthe membrane 1 is expanded to a proper size even the indentation at thelocation of the navel is reproduced, if it was somewhat hidden withinthe pear to start with. The membrane 1 would not show perfect detail.Nature imposed more functions than the membrane 1 would be able to show.But the overall representation would be close and appear “child like”.In the case of the elephant the membrane would produce one with only twolegs, but nature required the real one to be able to walk so it hasfour. The topic and use of constraints 4 is next. However constraints 4could be added to the bottom of the legs on the plot-frame 2 to give itflat pads. The more proper constraints 4 added the more close it wouldbecome in appearance to the real thing. One should even be able toreproduce some of the features that are in the head and joints of theelephant if care is taken in the formation and placement of them.

FIG. 8 shows the same plot-frame 2 and base 3 as FIG. 2 did with someconstraints 4 a and 4 b applied to the base 3 and within the plot-frame2. In FIG. 8 the membrane 1 is removed for clarity. Constraint 4 a couldbe made from a bent rod, or from sheet or block material as with 4 b. Itcould also happen that each constraint 4A and 4B are just two polesending at the proper locations, that of the top outside corners of theones shown in the FIG. 8. Constraints are made to hold the membrane 1 toany point/line/plane required, for the function intended. As just notedthere can be more than one that will satisfy the same condition andimpose the same function on the membrane 1. Constraints 4 can be made inany fashion that they need, as long as they hold the membrane 1 to therequired placement, and need not be in place until the membrane 1 ispartly of fully inflated; aiding in keeping an even surface tension uponthe membrane 1. They can be of any shape and on any area or side of themembrane 1, also could be of any number upon the membrane, but can notcover the whole surface of same. They could be made in sets so they holdthe membrane on both sides, act only to confine that point/line/plane ofthe membrane 1 to a required location for providing some function. Afunction on the membrane 1 could be for a door, connection plane toallow for a dome to be added to an existing wall that the dome needs toteam up with. A function on the membrane 1 acts only on the surface ofit, and because of that only on the surface of the structure itself.They can hold a portion of the membrane 1 flat and vertical to mountsome object to the shell of the structure that wouldn't work well if thesurface of it had a continuous curve. One such object might be a set ofcabinets. Constraints need not be in contact with the edge of theplot-frame 2. One such constraint 4 could be a match set used for a setof solar collection panels, and be held above and beneath the membrane1.

I have allowed the ones in FIG. 8 to be used for two different sizeddoors, or connection planes. Each of the constraints in FIG. 8 could beat any location, and the membrane 1 would still show all least surfacearea solutions. All shapes that the membrane 1 takes on would beprimary-geodesic shapes. Any polyhedral representation of the membrane 1would be a secondary-geodesic.

FIGS. 9 a and 9 b show the final shape of the membrane 1 that would beproduced in FIG. 8. FIG. 9 c is one of any secondary-geodesic polyhedralshapes that could be copied to the primary-geodesic surface, much in thesame manner that all past geodesic artworks have been copied to thesphere. It was done with the membrane 1 expanded or displaced from thesquare plot-frame 2 above and herein. Because the constraints in FIG. 8apply a fixed height to part of the surface of the membrane 1, they needto be to scale for the plot-frame 2 and target height and size of thefinal structure, and of the intended doors. There may or may not besimilar considerations for constraints intended for other functions ondifferent plot-frames 2.

FIG. 10 shows a simple arched rod or bent wire type of constraint 4 anda symmetrically cut plot-frame 2 for the membrane 1 to expand from. Oneor both halves of the surface of this shape of the membrane 1 as seen inFIG. 11 may be constructed at any one time, as the finances of theperson doing the work would allow. This adds a freedom to theengineering, manufacture and finance sides of the structure. As was thecase before, there could be constraints 4 placed around the edge of theplot-frame 2 for entryways. The option to add and remove constraints 4to FIG. 11 is endless. For the first time the primary geodesic surfacecan take upon it the wishes of the person that will have it in the end.There is no longer a need to confine geodesic surfaces to the sphere andthe rigidness of assumed rules of math, which has been used to give thepolyhedral surface.

FIG. 12 shows what can be done with repetitive use of just a square andrectangular plot-frame 2 and an expanded membrane 1. This clusterarrangement could also be made as the finances allow. Constraints 4would be made to match the tunnel pathways that run between the domes,as well as the intended size of the doors. Locations of the tunnels orpathways and the doors are open to the whims of the designer. Each unitin FIG. 12 would have a least area, for the functions imposed on themembrane 1. It could be that the plot of ground that is covered by thecluster could be cut into a single plot-frame 2 and the least surfaceand primary-geodesic surface could be found for that system as a whole.The membrane 1 expanded from that plot-frame 2 would be more flowing andless blocked than the one seen in FIG. 11. It would appear blocked onlywhere it was constrained for some intended function such as entryways.

The idea of primary-geodesic shapes allows a structure to take advantageof the functions that people require and the cost of maintainingcontrolled environments. The advantage of having the most volume ofspace inside a structure could come in handy when the outsideenvironment would tend to “bleed away” the air inside; such as would befound on Mars or the Moon. This system however is most creative whenused to cover the foundation or pad left behind from a structure thatwas removed by wind, fire, or water. Than edge of the pad or foundationwould than act as the plot-frame 2, and the base 3, with constraints 4in place at the required time during the final display of the membrane1. This would allow one to at least create a solid temporary cover intimes of need. There are a number of means to fabricate both the primaryand secondary geodesic surfaces found with this system. This artwork isnot intended to cover such assembly of the structure as there is alreadymany means available.

I. I claim a least surface geodesic shaped structure that matches up toany ground-plot and has the required polygonal functions such asconnection-planes, entryways, view-ports and other flat or curvedpredetermined surfaces that could be demanded upon its surface by,fixing the shape of any plot of ground that the structure is required tocover to an open window within a plot-frame (2); expanding the membrane(1) from the open window within the plot-frame (2) with a even pressureforce between the plot-frame and the base (3); while creating functionson the membrane (1) as polygonal surfaces whereby, constraints (4) holdthe membrane (1) to points/lines/planes by fixing the surface of themembrane (1) to the places that it needs to be for the polygonal shapesthat it is required to have, so that the membrane (1) shows the leastsurface above the plot-frame (2) with the proper shapes for thefunctions that the structure is required to have.